Generalized GRAPPA Operators for Wider Spiral Bands:Rapid Self-Calibrated Parallel Reconstruction forVariable Density Spiral MRI

Wei Lin,1* Peter Bornert,2 Feng Huang,1 George R. Duensing,1 and Arne Reykowski1

A rapid and self-calibrated parallel imaging reconstruction

method is proposed for undersampled variable density spiral

datasets. A set of generalized GRAPPA for wider readout line

operators are used to expand each acquired spiral line into a

wider spiral band, therefore fulfilling Nyquist sampling criterion

throughout the k-space. The calibration of generalized GRAPPA

for wider readout line operators is performed using the fully

sampled central k-space region. The resulting generalized

GRAPPA for wider readout line operator weights are adaptively

regularized to minimize the error in the newly-generated data at

different k-space locations. Simulation and experimental results

demonstrate that the technique can be used either to achieve a

significant acceleration and/or to reduce off-resonance artifacts

due to a shorten readout duration. Magn Reson Med 66:1067–

1078, 2011. VC 2011 Wiley-Liss, Inc.

Key words: variable density spiral; parallel imaging; GRAPPA;off-resonance; regularization; generalized GRAPPA for widerreadout line

Spiral magnetic resonance imaging (MRI) (1) has beensuccessfully applied to several important applications,including cardiac imaging (2), functional brain MRI (3),spectroscopic imaging (4), and motion correction (5). Allthese techniques benefit from the ability of the spiral tra-jectory to sample k-space rapidly using either single ormultishot acquisitions. A major challenge facing spiralMRI, however, is the tradeoff between the desire for ahigher imaging speed, which favors a smaller number ofspiral interleaves, and the need to avoid potentiallysevere off-resonance artifacts due to a longer readoutwindow.

Variable density spiral, where the central k-space is

sampled more densely than outer k-space, was first pro-

posed to increase the temporal resolution of dynamic

imaging (6), similar to the Cartesian key-hole imaging

method (7). Later, variable density spiral was also shown

to reduce aliasing (8), motion (9,10) and off-resonance

artifacts (11), again exploiting oversampling near the k-

space center. Alternatively, if the central k-space is

sampled at the Nyquist criterion while the outer k-space

is undersampled, variable density spiral will have a

shorter readout window length than a conventional uni-

form-density spiral. This results in reduced susceptibil-

ity artifacts, which has been exploited for coronary imag-

ing (12). However, the degree of undersampling at the

outer k-space is limited due to the concomitant loss of

image resolution and residual aliasing artifacts. In many

applications, it is desirable to further reduce the readout

duration and/or to further increase the imaging speed

without a sacrifice in the image quality.

When spiral data are acquired using a phased-array

coil, partial parallel imaging methods can be used either

to achieve acceleration by reducing the number of inter-

leaves, or to alleviate off-resonance artifacts by shorten-

ing the k-space trajectory and acquisition window for

each interleaf. General purpose parallel imaging recon-

struction techniques for non-Cartesian datasets have

been proposed and applied to spiral MRI, including the

iterative conjugate gradient sensitivity encoding (CG-

SENSE) (13,14) and parallel MRI with adaptive radius in

k-space (PARS) (15,16). Aliasing artifacts resulting from

undersampling can be significantly reduced with these

methods. However, the application of these techniques is

limited by long reconstruction time and a dependence

on sensitivity map measurements. More recently, several

more advanced partial parallel imaging methods have

been proposed for spiral imaging, mostly based on the

generalized auto-calibrating partial parallel acquisition

(GRAPPA) formalism (17–20). However, application of

these methods to variable density spiral datasets where

the distances between adjacent spiral lines vary continu-

ously across the k-space has not been reported in the

literature.In this work, a rapid k-space -based parallel imaging

reconstruction method is proposed for variable-densityspiral MRI. The technique is based on the idea to expandeach spiral line into a wider band, using a set of general-ized GRAPPA for wider readout line (GROWL) operators,therefore eliminating undersampled k-space regions. Thewidth of the spiral band can be adjusted in a flexiblemanner across k-space depending on the need to fulfillthe Nyquist sampling criterion. The calibration of theGROWL operators along various directions is performedusing the fully-sampled central k-space region. In addi-tion to being applicable to variable density spiral,GROWL provides two other advantages over existingmethods: First, it is completely self-calibrated, whicheliminates the need for any extra sensitivity information.This feature makes the method immune to possible arti-facts caused by motion between the sensitivity scan andthe accelerated scan. And second, GROWL is a veryrapid method when compared to many existing techni-ques, due to its noniterative nature, the small number ofGROWL operators and a small kernel size. A detailed

1Invivo Corporation, Philips Healthcare, Gainesville, Florida, USA.2Philips Research Europe, Hamburg, Germany.

*Correspondence to: Wei Lin, Ph.D., Invivo Corporation, Philips Healthcare,3545 SW 47th Ave., Gainesville, FL 32608. E-mail: wei.lin2@philips.com

Received 15 September 2010; revised 13 December 2010; accepted 8February 2011.

DOI 10.1002/mrm.22900Published online 5 April 2011 in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 66:1067–1078 (2011)

VC 2011 Wiley-Liss, Inc. 1067

description of the method is provided first, followed byan evaluation in both phantom and in vivo experiments.

MATERIALS AND METHODS

Variable Density Spiral Trajectory

A spiral trajectory can be described analytically in k-space as

kðuÞ ¼ AkrðuÞeiu; ½1�

where k-space radius kr is a monotonically increasingfunction of phase angle u. For a uniform-density spiral,kr(u) is proportional to u throughout the k-space. Forvariable-density spiral used in this work,

dkrdu

¼1 when u � u1

1þ N�1u2�u1

ðu� u1Þ when u1 < u � u2

N when u > u2

8<: ; ½2�

where N is the undersampling factor at outer k-spacewith phase angle larger than u2, with the sampling den-sity (du/dkr) decreasing smoothly from 1 to 1/N in thetransition zone between u1 and u2. The continuity ofdkr/du is necessary to ensure the continuity of gradientwaveform

GðtÞ ¼ 1

g

dk

dt¼ A

g

dudt

dkrdu

þ ikr

� �eiu ½3�

In this work, the function u(t) is defined as follows (21):

uðtÞ ¼ vt=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ at=Ta

p; ½4�

where Ta is the acquisition window length, a is a param-eter that controls the speed of transition from the initialgradient slew-rate limited regime to the later gradientamplitude limited regime, and

v ¼ umax

ffiffiffiffiffiffiffiffiffiffiffiffi1þ a

p=Ta: ½5�

The analytical approach presented here for spiral gradi-

ent waveform design is a compromise between the com-

putational burden and a reasonably compact gradient

waveform. Although it is not within the scope of this

work to discuss the compactness of our gradient wave-

form, our simulation shows that the total acquisition

window with our method is prolonged no more than

12% than that using a much more computational inten-

sive numerical design method (similar to Ref. 22) to min-

imize the gradient duration.

Figure 1 shows the sampling density (Fig. 1a) and k-

space trajectories (Fig. 1b–d) of a uniform-density and a

variable-density spiral, both with a readout duration of

Ta ¼ 6 ms. The uniform-density spiral was designed to

fully sample k-space with 12 interleaves. Variable-den-

sity spiral was designed to fully sample the central k-

space region with 4 interleaves. After the application of

GROWL operators, the Nyquist sampling criteria can be

fulfilled through the k-space, therefore providing an

acceleration factor of R ¼ 3.

There are three parameters that define an under-sampled variable density spiral trajectory: the radius ofcentral fully-sampled circle kr0, the width of transitionregion Dkr, and the undersampling factor N at outer k-space. These parameters (kr0, Dkr, N) can be adjusteddepending on the desired application and the need tominimize reconstruction error, as will be discussed next.

Generalized GRAPPA for Wider Readout Line (GROWL)Operator

Previously, a k-space based parallel imaging operator

with multiple k-space source points along the readout

direction has been proposed (23) and applied to azimu-

thally under-sampled radial datasets, by expanding each

radial line into a wider radial band. In this work, this

operator is further generalized and applied to spiral data-

sets. Each spiral line is expanded into a wider spiral

band consisting of multiple lines parallel to the origi-

nally acquired spiral lines (Fig. 2). The width of the

expanded spiral band can vary across the k-space, as

determined by the need to fulfill the Nyquist criterion.

As the GROWL operator slides along the spiral trajectory,

target k-space points (open circles in Fig. 2b) are com-

puted from a weighted sum of acquired source k-space

points (gray circles in Fig. 2b), exploiting the implicit

spatial encoding embedded in the multicoil dataset.The calibration of the GROWL operator is performed

using the fully sampled central k-space region (centralcircle in the Fig. 2a). For each desired GROWL operator

FIG. 1. Design of a variable density spiral trajectory. a: The sam-

pling density of a uniform density (UD), shown by the dotted line,and a variable density (VD) spiral, shown by the solid line. b: k-space trajectory of one interleaf of both UD and VD spiral, each

with readout window of Ta ¼ 6 ms. c, d: k-space trajectories of 12UD spiral interleaves (c), fulfilling the Nyquist sampling theorem

everywhere, and 4 VD spiral interleaves, fulfilling Nyquist only inthe central part (d).

1068 Lin et al.

orientation, fully-sampled central region was firstregridded onto a rotated Cartesian grid (24), followed bythe calibration of the GROWL weights on this grid. Notethat since the Nyquist criterion is already fulfilled inthis region, there is no need to generate new data here.

There are three parameters that determine a GROWL

operator: the number of source points (Nx), the distance

of target points from the acquired spiral line (Dky), andthe orientation of the relative shift y (cf. Fig. 2b). The

first two parameters have been studied in detail using

undersampled radial datasets in Ref. 23. More source

points result in more accurate estimation at a cost of a

higher computational burden, while both estimation

error and noise amplification increase for target points

further away from the acquired readout points. For an

eight-channel circular coil, 3-5 source points (with a

spacing of 1/FOV) are sufficient to achieve a balance

between accuracy and speed, while target points with a

Dky no more than 2–3/FOV can be estimated to a high

accuracy, depending on the data noise level (23). For the

third parameter, spiral datasets present an additional

challenge, since orientation of the relative shift y can

vary continuously between 0 and 2p, unlike the discrete

set of angles in a radial dataset. However, it was previ-

ously shown that a GRAPPA kernel with arbitrary orien-

tation can be approximated by interpolating between two

kernels with similar orientations (16,18). Therefore, a set

of basis GROWL operators with different orientations

can be calibrated, followed by a simple linear interpola-

tion to determine the operator weights for an arbitrary

direction of relative shift y.An additional source of GROWL operator error comes

from the curvature of the spiral sampling trajectory. As aGROWL operator is calibrated using source points on astraight line, but applied to a curved readout line, someadditional error will be introduced. For variable densityspiral, this error can be limited by choosing a sufficientlylarge radius (kr0) for the fully-sampled k-space center,where GROWL operators is not applied (cf. Fig. 2a), butthe curvature along the readout line is the largest.

K-Space Adaptive Tikhonov Regularization

An optimal Tikhonov regularization factor was previ-ously proposed to minimize the fitting errors for k-space-based parallel imaging methods such as GROWL andGRAPPA (23,25). To recap, if each GROWL operator ker-nel is calibrated using a source data matrix S and a targetdata vector t in the auto-calibration signal region, thenthe optimal GROWL operator weight vector wopt withTikhonov regularization is:

wopt ¼Xnj¼1

sj

s2j þ l2uHj tvj ½6�

Here uj, vj, sj are the left, right singular vectors and sin-gular values of source data matrix S respectively, aftersingular value decomposition. The parameter l repre-sents the Tikhonov regularization factor.

If a single Tikhonov factor was applied throughout thek-space, Ref. 23 showed that the optimal global Tikho-nov factor l0 is approximately

l0 �ffiffiffiffiffiffiffiNE

ps: ½7�

Here it is assumed that each channel of MR data con-tains independent gaussian noise with a standard devia-tion of s and the entire k-space contains NE data points.

Similar regularization approaches based on analyzingthe singular values of the reconstruction equation andestimating noise contribution have been proposed previ-ously and were applied to both SENSE (26) andGRAPPA (27). When compared with earlier strategiessuch as the L-curve method (28), such an approach ismore robust and gives superior results (26,27).

In this work, we investigate whether the Tikhonov regu-larization factor can be adjusted locally across the k-spaceto further minimize data estimation error. In other words,the ratio between the optimal local Tikhonov factor loptand l0 becomes a function of the k-space radius kr:

RðkrÞ ¼ lopt=l0 ½8�

The shape of R(kr) will be investigated using a simulateddataset in the next subsection.

GROWL Operator Error Simulation Study

The appropriate parameters to minimize GROWL opera-tor errors from different sources were investigated sys-tematically using a simulated dataset. A noise-free T1-weighted Cartesian brain MR image (matrix size 256 �256) was downloaded from a simulated brain database(http://www.bic.mni.mcgill.ca/brainweb/). The complexsensitivity profile of a head coil array with 8 coil ele-ments equally spaced around a cylinder was computedusing an analytic Biot-Savart integration. The k-spacedata along Cartesian and spiral trajectories for each indi-vidual channel were then derived using the Fouriertransform and inverse gridding. Gaussian distributed ran-dom noise was added to both real and imaginary compo-nents of each channel, resulting in a noise standard devi-ation in the range of 0.2–5.0% of signal intensity of the

FIG. 2. Basic principle of GROWL operator. a: For a variable den-sity spiral dataset, GROWL operators are used to expand each

spiral line into a wider band, estimating the missing k-space data.The width of the band depends on the need to fulfill the Nyquist

criterion. Fully-sampled region in the center of k-space (the centralcircle) is used for the calibration of GROWL operators. b: Thethree parameters that determine a GROWL operator kernel are

illustrated: The number of source points Nx, the distance of thetarget point from the readout line Dky and the orientation of theoperator y.

Generalized GRAPPA Operator for Wider Spiral Bands (GROWL) 1069

white matter (the dominant tissue) in the final imagesreconstructed using the square-root-of-sum-of-squarechannel combination.

To investigate the effect of angular interpolation onGROWL operator errors, the k-space was rotated to 256orientations using a shearing method (24). The GROWLoperator weights were then computed using the central48 � 48 matrix in the k-space for all 256 orientations.The interpolated GROWL weights were generated using8, 16, 32, 64 and 128 basis orientations and comparedwith the gold-standard weights for all 256 orientations,using the normalized root-mean-squared-error (RMSE).

To investigate GROWL operator error introduced bythe curvature of the readout lines, source data points onboth straight and curved readout lines were computed.For the latter, different curvatures were assumed, corre-sponding to the different k-space radius kr of a spiral tra-jectory. Different k-space locations were examined, rang-ing from near the center (kr ¼ 0.1 kmax

r ) where thecurvature of the spiral line is greatest, to the peripheral(kr ¼ kmax

r ) where the curvature is smallest. The GROWLweights calibrated using straight lines were applied tosource k-space points along both linear and curved read-out lines to derive target k-space points, and the addi-tional curvature-induced errors were quantified.

To investigate k-space adaptive regularization, GROWLoperator fitting errors were evaluated at k-space locationsranging from kr ¼ 0.1 kmax

r � kmaxr , with the Tikhonov

factor l ranging from 0.1 to 1000 times of l0 (cf. Eq. 7).The optimal Tikhonov factor lopt that minimizes thelocal k-space fitting error was then determined and com-pared with global optimal Tikhonov factor l0. Theamount of fitting error was compared between adaptiveand fixed regularization, for different kr (k-space radius)and Dky values (the distance between the target pointand the source readout line). GROWL operator weightscorresponding to different regularization factors werealso compared.

GROWL Reconstruction and Imaging Experiments

Based on results derived from the simulation study, sev-eral undersampled variable density spiral trajectories weredesigned which allow accurate GROWL reconstruction atdifferent acceleration factors. The central fully-sampled k-space region kr0 was set to 0.2 kmax

r . The transition region’swidth Dkr was varied from 0.1 to 0.3 kmax

r . The outer k-space region was undersampled by the factor N rangingfrom 2.4 to 5.6. The resulting net acceleration factors are R¼ 2.0 – 4.0, as determined by the number of spiral inter-leaves required when compared with the uniform densityspiral with an identical acquisition window length.

For the GROWL reconstruction, 32 basis directions (inthe range of 0 to p) were used since our simulation resultsshow that they are enough to interpolate the GROWLweights accurately. GROWL weights in the angular rangeof p to 2p can be derived by simply rearranging the order-ing of source and target points in these 32 weight sets,therefore reducing the computation cost by half. For eachbasis direction, the central fully sampled region was usedto derive a set of singular values and vectors (cf. Eq. 6).Different Tikhonov regularization factors were then

applied to derive the optimal GROWL operator weights,depending on k-space radius. For actual imaging experi-ments, since the acquired spiral data will have variablespacing along each spiral line, data was first sampled ontoequally-spaced k-space points using linear interpolation,before their convolution with the GROWL weights. Finallythe data was compensated for the sampling density (29)before regridding (29,30).

The performance of GROWL reconstruction at differentacceleration factors and data noise levels were evaluatedfirst using simulated spiral datasets. Those results werecompared with direct regridding and the GRAPPA Oper-ator Gridding (GROG) (31) followed by the conjugate-gra-dient SENSE reconstruction (32) (GROG þ CG-SENSE).The noise-free brain dataset previously used in the simu-lation study was appropriately weighted with the receivecoil sensitivities and inverse regridded to generate vari-able density spiral datasets with additive noise levelsranging from 0.2 to 5.0%. Errors of the correspondingreconstructions were evaluated using RMSE and the vis-ual assessment of difference image from the gold-stand-ard reference image.

The application of GROWL reconstruction to under-sampled variable density spiral datasets was then exam-ined in phantom and in vivo brain experiments, using a3.0T clinical scanner (Achieva, Philips, Best, The Neth-erlands) equipped with an eight-channel head coil array(Invivo, Gainesville, FL). In these experiments, multi-slice 2D T2-weighted gradient echo imaging was used toacquire both fully-sampled uniform density and under-sampled variable density spiral datasets with differentacceleration factors. The common scan parametersinclude: FOV 230 � 230 mm2, slice thickness 5mm, ma-trix size 256 � 256, TR/TE ¼ 500/16 ms, flip angle ¼18�. In the phantom experiment, a off-resonance fre-quency (DB0) map was also acquired, allowing the cor-rection of the off-resonance artifact using the conjugatephase method for each channel (33), followed by sum-of-square reconstruction. For the in vivo brain scans, aspectrally selective saturation pulse precedes the excita-tion pulse to suppress the fat signal.

To demonstrate the acceleration achieved with vari-able density spiral sampling and GROWL reconstruction,multiple pairs of uniform and variable density spiraldatasets were acquired with identical acquisition win-dow length Ta ¼ 4.4, 5, and 7 ms. However, uniformdensity datasets contains 32, 48, and 64 spiral inter-leaves, while variable density spiral datasets werealways acquired with 16 interleaves, corresponding toacceleration factors of R ¼ 2, 3, and 4. To demonstratethe reduction of off-resonance artifact, a uniform and avariable density spiral datasets were acquired with 64interleaves in the phantom experiment, with an acquisi-tion window of Ta ¼ 6 ms and 2.1 ms, respectively. Todemonstrate the utility of the proposed method for fast,motion-free and self-calibrated acquisition for in vivoscans, a single-shot variable density spiral dataset wasacquired using an acquisition window of Ta ¼ 20 ms toachieve an image resolution of 96 � 96. For comparison,a single-shot uniform density spiral dataset with an ac-quisition window of Ta ¼ 50 ms was also acquired toachieve the identical image resolution. In addition, a

1070 Lin et al.

high-resolution (256 � 256) Cartesian reference imagewas also acquired using a scan time of 2 mins.

RESULTS

Simulations

Figure 3 shows simulation results which studied the twounique aspects of the GROWL operator when applied to

spiral data: angular interpolation of the GROWL weightvectors and the effect of curved readout lines. Figure 3ashows that weight vectors are more accurate when morebasis orientations are used for linear interpolation. Fornoise levels in the range of 0.2–5.0%, 32 basis orienta-tions were found to be sufficient to reduce the contribu-tion of the angular interpolation to less than 5% root-mean-square-error (RMSE) for the GROWL weight

FIG. 3. GROWL operator estimation errors. a: RMSE in the GROWL operator weights when using different number of basis orientationsfor angular interpolation (s indicates different noise levels). b: Additional k-space fitting error introduced by curved vs. straight readout

lines, at different k-space radius kr.

FIG. 4. Adaptive Tikhonov regularization. a: The optimal regularization factor lopt, as determined by minimizing the GROWL operator fit-ting error, at different k-space locations. b: Normalized fitting error, with adaptive (lines with markers) and fixed (lines without markers)regularization, at different k-space locations. c: Normalized fitting error with adaptive regularization, for GROWL operators with two dif-

ferent distances between target points and source readout lines: Dky ¼ 1/FOV (lines with markers) and Dky ¼ 2/FOV (lines withoutmarkers). d: The relationship between the maximal magnitude of GROWL weight vectors and the regularization factor l.

Generalized GRAPPA Operator for Wider Spiral Bands (GROWL) 1071

vectors. Therefore 32 basis orientations were used for theremainder of this work to achieve a balance between ac-curacy and reconstruction speed. Figure 3b shows thedifference in normalized k-space fitting error whenGROWL operators were applied to curved vs. straightreadout lines. The curvature of a spiral readout line isapproximately inversely proportional to the k-space ra-dius kr.It can be seen that the curvature-induced error ishighest near k-space center, where the curvature is thelargest, and drops rapidly as kr increases. For all noiselevels, curvature-induced k-space fitting error is lessthan 5% when kr > 0.2 kmax

r . As a result, all variabledensity spiral trajectories used in this work has a centralfully sampled region with a kr0 ¼ 0.2 kmax

r to limit thecurvature-induced error.

Figure 4 shows simulation results for the proposedadaptive Tikhonov regularization factor to minimize theGROWL operator error at various k-space locations. Fig-ure 4a shows the optimal lopt that minimizes the localGROWL operator fitting error at different k-space loca-tions. A nearly linear relationship exists between k-spaceradius kr and lopt in the logarithmic scale:

RðkrÞ � lopt=l0 � S log ðkrÞ ½9�Hereafter S is referred as the slope of Tikhonov regulariza-tion. The l0 originally proposed in Ref. 23 achieved nearlyoptimal regularization near the k-space center, while larger

regularization factors were needed in the outer k-space.This relationship is independent of the data noise level.As shown in Fig. 4b, adaptive regularization (when loptwas used across k-space) significantly reduces the GROWLoperator error at the outer k-space, when compared tofixed regularization (when l0 was used across k-space).This pattern is again consistent for datasets with differentnoise levels. Note that the difference between adaptiveand constant regularization is very small near k-space cen-ter. This shows that the l0 proposed in Ref. 23 is valid ifthe goal is to minimize the overall error using a single reg-ularization factor, since the majority of image energy con-centrates near the k-space center. Figure 4c furthercompares the GROWL operator error for target points atDky ¼ 1/FOV vs. 2/FOV, both after adaptive regularization.It can be seen that the error is much higher for Dky ¼ 2/FOV. Figure 4d shows a nearly piece-wise linear relation-ship between the maximal magnitudes for GROWL opera-tor weight vectors and lopt in a logarithmic scale. Asimilar relationship was observed for each component ofthe GROWL weight vector. Combining results from Fig.4a,c a nearly linear relationship exists between the optimalGROWL operator and k-space radii. Therefore, several setsof GROWL weights can be generated that minimize theerror at different k-space radii, and a simple linear interpo-lation can be used to generate optimal weights for any k-space position.

FIG. 5. Comparison of regrid-

ding, GROWL and GROG þ CG-SENSE for variable-density spiral

data. a: Simulated fully-sampledreference image. b–g: Recon-struction results of simulated

variable density spiral with anoverall data reduction factor of R

¼ 3, at input noise levels of1.0% (b, d, f) and 5.0% (c, e, g):(b, c) Regridded images, (d, e)

GROG þ CG-SENSE results, (f,g) GROWL reconstructions. TheRMSE of each reconstruction is

shown in the upper right cornerof each panel.

1072 Lin et al.

Figure 5 compares the GROWL reconstruction withregridding and GROG þ CG-SENSE using a simulatedvariable density spiral dataset with two added noise lev-els (1.0 and 5.0%). This dataset has a fully sampled cen-tral k-space region with a radius of kr0 ¼ 0.2 kmax

r , a tran-sition width of Dkr ¼ 0.3 kmax

r , and an outerundersampling factor of N ¼ 4.0, resulting in a net accel-eration factor of R ¼ 3. As shown in Fig. 5b,c, directregridding result in severe image blurring and residualaliasing artifacts. The GROG þ CG-SENSE reconstructiononly slightly improve image quality over the directregridding (Fig. 5d–e). In contrast, the GROWL recon-struction significantly improved the image resolutionand reduced aliasing (Fig. 5f,g). Among three reconstruc-tion schemes, GROWL yields the lowest RMSE at bothnoise levels.

The importance and flexibility of the proposed adapt-ive regularization strategy is demonstrated in Fig. 6,using a simulated dataset with an added noise level of5.0%. Direct regridding (Fig. 6a) results in severe imageblurring and aliasing artifacts due to undersampling inthe outer k-space (Fig. 6e). GROWL reconstruction witha fixed regularization (Fig. 6b) removes blurring and ali-asing, but suffer from significant noise amplification(RMSE ¼ 8.0%). The k-space signal magnitude (Fig. 6f)shows an over estimation of outer k-space data points,which is a result of significant GROWL operator errorsthere. When the proposed adaptive regularizationscheme is used (Fig. 6c), noise is significantly reduced(RMSE ¼ 4.0%) due to the suppression of noise amplifi-cation in the outer k-space (Fig. 6g). However, residualaliasing artifact may appear quite unpleasant for someobservers. Fortunately, a smaller slope of Tikhonov regu-

larization S can be used to reduce the amount of regula-rization, resulting in a compromise between the noiseand artifact levels (Fig. 6d, RMSE ¼ 6.0%). The resultingk-space signal magnitude (Fig. 6h) in the outer k-space isat an intermediate level between the fixed regularization(Fig. 6f) and the adaptive regularization (Fig. 6g) thatminimizes the RMSE.

Table 1 summarizes the RMSE of GROWL reconstruc-tion using the simulated datasets at different noise leveland acceleration factors, when the proposed adaptiveregularization scheme is applied. As expected, RMSEscales with noise level and acceleration factors. Overall,GROWL reconstruction achieves a very low RMSE, closeto the noise floor. In one case (i.e., s ¼ 5.0%, R ¼ 3),RMSE is even lower than the noise floor.

Phantom and In Vivo Results

Figure 7 shows images from a phantom imaging experi-ment, demonstrating the reduction of off-resonance arti-facts with variable density spiral and GROWL

FIG. 6. Tradeoff between noise and artifacts. a: Regridded image using a simulated undersampled variable density spiral data (R ¼ 3)

with an added noise level of 5.0%. b–d: GROWL images with a (b) fixed regularization, (c) an adaptive regularization to minimize RMSE,and (d) a modified adaptive regularization to achieve a balance between noise and residual artifact level. RMSE of each reconstructionis shown in the upper right corner of each panel. e–h: Corresponding k-space magnitude representations are shown for one of the eight

channels (logarithmic scale).

Table 1Root Mean Squared Error (RMSE) of GROWL Reconstruction withAdaptive Regularization at Different Acceleration Factors, Using

Simulated Variable Density Spiral Datasets with Different AddedNoise Levels

Input datanoise level s (%)

Net acceleration factor R

2.0 3.0 4.0

0.2 1.1% 1.3% 2.1%1.0 1.6% 2.0% 3.1%5.0 3.7% 4.0% 5.2%

Generalized GRAPPA Operator for Wider Spiral Bands (GROWL) 1073

reconstruction. Figure 7a shows the image reconstructedfrom a uniform density spiral dataset with an acquisitionwindow Ta ¼ 6 ms. Field inhomogeneity causes blurringartifacts, particularly towards the edge of the phantom.Variable density spiral was used to shorten Ta to 2.1 ms,which removes most blurring artifacts (Fig. 7b). When astandard gridding reconstruction was used, some aliasingartifacts can be seen, due to undersampling at outer k-space. With the GROWL reconstruction using the samedataset, residual aliasing artifacts can be furtherremoved.

Figure 8 shows the usage of variable density spiralsampling and GROWL reconstruction for scan timereduction in a phantom experiment. The images shownin Fig. 8a,b were reconstructed from a uniform densityspiral dataset with 64 interleaves using a fixed readoutwindow of Ta ¼ 6 ms, before (Fig. 8a) and after (Fig. 8b)off-resonance correction using the measured DB0 map. Ifonly 16 interleaves are used for image reconstruction (re-gridding), undersampling causes severe aliasing artifacts(Fig. 8c). A variable density spiral trajectory with the

same readout duration, however, was able to fully sam-ple central k-space with only 16 interleaves, achievingan acceleration factor of R ¼ 4. While direct regriddingcauses severe residual aliasing artifact (Fig. 8d) due toundersampling in the outer k-space, GROWL reconstruc-tion results in an aliasing-free image (Fig. 8e). Due to thestrong B0 inhomogeneity, some blurring artifacts remain.After a conjugate phase correction using the DB0 map, ar-tifact-free image can be generated (Fig. 8f), similar toresults achieved using the fully-sampled uniform densityspiral datasets (Fig. 8b).

Figure 9 shows results from a high-resolution in vivobrain scan. Figure 9a shows the reference fully-sampleduniform density spiral image. Figure 9b–d show GROWLreconstructions using variable density spiral datasetswith acceleration factor of R ¼ 2, 3 and 4. GROWL wasable to significantly improve the image resolution, andremove most aliasing artifacts when compared with thedirect regridding (Fig. 9e–f). As expected, more noiseand residual artifacts appear in the GROWL images asthe acceleration factor increases.

FIG. 7. Off-resonance artifact reduction. a: Image from a fully-sampled uniform density spiral. b, c: Images from an undersampled (R ¼4) variable density spiral data reconstructed with regridding (b) and GROWL (c). All datasets were measured and contains 64 spiral inter-

leaves. Uniform density spiral has a readout window of Ta ¼ 6 ms, while variable density spiral allowed for a readout window of Ta ¼2.1 ms. No off-resonance correction was performed.

FIG. 8. Scan time reduction.Using variable density spiral

sampling and parallel reception,scan time can be reduced asdemonstrated in a phantom

experiment. a, b: Images from afully-sampled uniform density

spiral dataset (64 interleaves),before (a) and after (b) applyingconjugate phase correction. c:Reduction factor R ¼ 4 uniformdensity spiral image (16 inter-

leaves). d–f: Images from a 16-interleaf variable density spiraldataset, using regridding (d),

GROWL (e), and with conjugatephase correction (f). The readout

windows for all datasets are Ta¼ 6 ms.

1074 Lin et al.

Figure 10 demonstrates a potential single-shot applica-tion of the proposed self-calibrated, variable spiral paral-lel imaging method, which can be used to reduce off-res-onance artifacts in vivo. When compared with a high-resolution 256 � 256 Cartesian reference image (Fig.10a), a single-shot uniform-density spiral dataset with animage matrix of 96 � 96 requires a readout window of50 ms, resulting in significant image blurring (Fig. 10b).In contrast, a variable density spiral dataset with anidentical image matrix only requires a readout windowof 20ms. Direct regridding results in a low-quality imagewith little anatomical information (Fig. 10c). WithGROWL reconstruction, image resolution is greatlyimproved (Fig. 10d), revealing many anatomical featuresthat can be seen in the high-resolution Cartesian scan.

DISCUSSIONS

In this work, a rapid self-calibrated parallel imagingreconstruction method is proposed for undersampledvariable density spiral datasets. Each spiral line is wid-ened into a band consisting of several parallel linesusing a set of GROWL operators. The width of the bandis flexibly adjusted based on the need to fulfill theNyquist sampling criterion at different k-space locations.Fully sampled central k-space region allows the self-cali-bration of the GROWL operators.

When used in conjunction with the GROWL recon-struction, undersampled variable density spiral acquisi-tion can be used either to achieve scan time reduction(with fewer interleaves), or to reduce off-resonance arti-fact due to a shorter readout duration (see Figs. 7 and10). On the other hand, all algorithms previously devel-oped for off-resonance correction (33–35) can be applied

to GROWL reconstruction results. Since GROWL opera-tor only use adjacent source points on the originallyacquired spiral readout lines, any off-resonance phase

FIG. 9. Multishot brain scans. Variable density spiral parallel imaging in combination with GROWL reconstruction demonstrates thescan time reduction. a: Reference image from a fully-sampled uniform density spiral scan (b–d) Undersampled variable density spiralscans reconstructed with GROWL with acceleration factor of R ¼ 2, 3, and 4, respectively. e–g: Regridding reconstruction using the

same datasets as those used in (b–d).

FIG. 10. Single-shot brain scans. Variable density spiral in combi-

nation with GROWL reconstruction demonstrate the potential forthe reduction of the acquisition window. a: High-resolution Carte-

sian reference image (scan time ¼ 2 mins, imaging matrix 256 �256). b: Single-shot uniform-density spiral (readout window Ta ¼50 ms, imaging matrix 96 � 96). c, d: Undersampled variable den-

sity spiral (readout window Ta ¼ 20ms, imaging matrix 96 � 96),with direct regridding (c) and GROWL reconstruction (d).

Generalized GRAPPA Operator for Wider Spiral Bands (GROWL) 1075

accumulated is passed on to the target data points,which lies near source points in k-space. This makes itfeasible to de-blur each individual coil image usingstandard methodology available.

The GROWL operator provides a very flexible approachto parallel imaging with arbitrary k-space trajectory, sinceonly several adjacent source data points along the readoutline are used to estimate target points at a distance Dkyfrom the acquired line. On one hand, this allows theGROWL operator to be applied to an arbitrary k-space tra-jectory, since curvature of any trajectory is always finitedue to a limited MR scanner gradient slew rate. On theother hand, the curvature of the source readout line is anintrinsic source of error for the GROWL operator. In thiswork, this problem is essentially circumvented by using asufficient large fully-sampled k-space center, since thecontribution of curved readout is negligible outside thiscircle. A possible future improvement is to computeGROWL operator weights for readout lines with variouscurvatures. For variable density spiral, this can result insmaller fully-sampled center and higher acceleration fac-tors, at a cost of longer computation times.

Comparison with Existing Techniques

In this work, we have compared GROWL with GROG þCG-SENSE, which is another technique that can beapplied to variable-density spiral datasets. At all noise lev-els and acceleration factors examined, GROWL generatesimages with lower RMSE and higher visual quality thanGROG þ CG-SENSE. A limitation of the GROG operator isthat it can only estimate k-space data points no fartherthan 0.5/FOV from acquired samples, to avoid excessiveestimation errors. In contrast, GROWL can estimate k-space data points within a 2/FOV – 3/FOV radius from theacquired sample points, which extends the k-space cover-age and provides better image reconstruction. On the otherhand, it is possible to combine GROWL with conjugate-gradient SENSE to further improve the reconstruction.

One major advantage of the GROWL operator, when com-pared with several existing parallel imaging techniques(13,15,17,20) which are also potentially applicable to vari-able density spiral, is a much lower computational cost dueto the non-iterative nature of the reconstruction, the smallkernel size, and the usage of a kernel interpolation scheme.The total reconstruction time is 5–10 s for each 256 � 256dataset on a 3.0GHz personal computer. The majority of thereconstruction time is spent on the calibration of GROWLoperators, while the application of GROWL operators andthe final regridding step only takes 100–300 ms for eachimage. In dynamic imaging applications, GROWL operatorand regridding weights can be saved and applied repeti-tively, therefore enabling real-time parallel reconstructionfor highly undersampled spiral data.

Achievable Acceleration Factors

There are three parameters that define an undersampledvariable density spiral trajectory and determine theachievable acceleration factor: the radius of central fully-sampled circle kr0, the width of transition region Dkr,and the undersampling factor N at outer k-space. The

choice for these parameters is mainly dictated by theneed to minimize the GROWL operator error. A suffi-ciently large area near the k-space center (kr0 > 0.2 kmax

r )should be fully-sampled to minimize the error intro-duced by the curved readout lines. The other two param-eters are mainly constrained by higher GROWL operatorerror when target points move further away from thesource readout lines. Figure 4b shows that GROWL oper-ator error is larger for Dky ¼ 2/FOV than Dky ¼ 1/FOV. Alarger undersampling factor N will result in highernoise/artifact level due to the need for a GROWL opera-tor with a larger Dky value, which is necessary to fulfillthe Nyquist criterion. Similarly, a smaller transitionwidth Dkr would move the application of GROWL opera-tors with large Dky values closer to the k-space center,which would increase the reconstruction error. Combin-ing these considerations, the maximal acceleration factorexamined is limited to 4 for the 8-channel head coilused in this work. However, recent developments ofhigh-channel receive coils (36) may provide opportuni-ties for higher acceleration.

Adaptive Tikhonov Regularization

A second contribution of this work is an adaptive Tikho-nov regularization factor to minimize the GROWL opera-tor throughout the entire k-space. It was found that opti-mal Tikhonov factor which minimizes the local k-spacefitting error changes gradually against the k-space radiuskr. The Tikhonov factor previously proposed in Ref. 23(l0 in Eq. 7) indeed performs well near the k-space cen-ter. Further away from the k-space center, however, ahigher l will result in a lower local k-space fitting error.As a result, the GROWL operator weights can beadjusted continuously across the k-space to furtherreduce the overall reconstruction error. This is particu-larly important for noisy data, where noise amplificationby the GROWL operator can be quite high towards theperipheral of the k-space.

When compared with the fixed regularization pro-posed in Ref. 23, the overall reconstruction error can sig-nificantly be reduced with the proposed adaptive regula-rization scheme (see Fig. 6). Table 1 shows that theproposed method can achieve very low error (RMSE). Ifthe noise is distributed uniformly in k-space, the lowestRMSE that can be achieved with any parallel imagingmethod is s

ffiffiffiffiR

p, where r is the noise standard deviation

and R is the acceleration factor. Table 1 show that thislimit was surpassed in many datasets with GROWL. Thisis mainly due to two reasons: First, although a variabledensity spiral trajectory under-samples a majority area ofthe k-space, there is a small circle near the k-space cen-ter that is oversampled, similar to the intrinsic oversam-pling near k-space center for an undersampled radialdataset. Since the majority of image energy concentratesnear the k-space center, the effective signal averagingnear k-space center will increase the signal to noise ratio(SNR). Second, when the noise level of input data isquite high (e.g., 5%), actual noise-free k-space data couldlie well below the noise floor towards the edge of the k-space. The proposed adaptive regularization scheme usesa very high regularization factor kopt at these regions.

1076 Lin et al.

This results in GRAPPA weight vectors with very smallmagnitudes (see Fig. 4d), effectively suppressing thenoise in the outer k-space. This does come with somesacrifice of image resolution, but will further increasethe SNR. Future work has to determine if the proposedadaptive regularization scheme can be applied to other(e.g., radial) datasets.

As demonstrated in Fig. 6, the proposed adaptive regu-larization scheme also allows users to adjust the balancebetween the noise and residual artifact level in thereconstructed images. This is achieved by adjusting theslope of regularization factor l across the k-space. Onthe other hand, a caveat is that an image with the lowestRMSE error is not necessary the one with the best imagequality. Figure 6c has a lower RMSE than Fig. 6d, butmany observers would find the image quality in Fig. 6dmore acceptable. Future independent observer studiesare necessary to determine whether such an adaptiveimage reconstruction approach can indeed have a posi-tive impact on the diagnostic quality of images.

Possible Extension

A possible extension of the current work is to use GROWLoperators to directly estimate target data points on a Carte-sian grid, therefore eliminating the final regridding proce-dure. Such an approach may provide a further computa-tional advantage over the procedure used in this work dueto two factors: First, there will be no need to compensatefor the sampling density of a non-Cartesian dataset. Sec-ond, there will be no need for final regridding and fastFourier transform can be directly applied for image recon-struction. To implement such a ‘‘direct-GROWL’’ scheme,each target Cartesian grid can be assigned to a source spi-ral line to which it has the shortest distance. To accountfor noninteger source-target distance, a series of GROWLoperator weights with different source-target distancescan be computed and properly interpolated, in anapproach similarly to one previously proposed for 1Dnon-Cartesian parallel imaging using GRAPPA (37).

In summary, the GROWL operator for wider spiralband provided two advantages when applied to anundersampled variable density spiral dataset. First, themethod is completely self-calibrated, requiring no addi-tional sensitivity information. Instead, the fully sampledcentral k-space region present in a variable density spiraldataset is used for the calibration of GROWL operatorsalong different directions. Second, GROWL is very com-putational efficient due to the noniterative nature, smallkernel size and small number of the kernels. Possibleapplications of the proposed technique include func-tional brain and diffusion MRI as well as applications inprospective motion correction.

ACKNOWLEDGMENTS

The authors thank Miha Fuderer for stimulatingdiscussions.

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